#143781

order	512
nilpotency class	4
isoclinism	isoclinic class of #143703
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

399 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 399 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(17,18,22,28)(19,30,33,44)(20,29,23,26)(21,25,37,42)(24,34,40,31)(27,35,43,36)(32,45,47,39)(38,46,41,48);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,19,22,33)(18,24,28,40)(20,34,23,31)(21,32,37,47)(25,41,42,38)(26,44,29,30)(27,39,43,45)(35,48,36,46);(18,25)(20,23)(21,36)(24,38)(26,43)(27,29)(28,42)(30,45)(31,34)(32,46)(35,37)(39,44)(40,41)(47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=g,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=dg,d^c=di,e^c=eh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50977);